# APPENDIX

 Chapter X - Maximum Performance Efficiency Approaches for Estimating Best Practice Costs Data Mining: Opportunities and Challenges by John Wang (ed) Idea Group Publishing 2003
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### Proof of Theorem 1:

1. Consider the potentially feasible solution m0 = k = a0r for all r. Then,

Thus the solution m0 = k = a0r is feasible for problem MPE (λ) for all λ > 0, for k0 = min (Σyrj)1 which is positive due to the positivity of the data. It follows that.

2. Here we note that the ar* (λ) for the MMPE model, while feasible, are not necessarily optimal for the original MPE model. Hence use of these values in the MPE objective function will generally give a lower objective function value.

### Proof of Theorem 2:

This is a modification of a proof for a version of the theorem given in Troutt (1993). By the assumption that x is uniformly distributed on {x:w(x) = u}, f(x) must be constant on these contours; so that f(x) = φ(w(x)) for some function, φ( ). Consider the probability P( u w(x) u + ε ) for a small positive number, ε. On the one hand, this probability is ε g(u) to a first order approximation. On the other hand, it is also given by

Therefore

Division by ε and passage to the limit as ε 0 yields the result.

### Further Details on the Simulation Experiment

To simulate observations within each data set, a uniform random number was used to choose between the degenerate and continuous portions in the density model

where p =0.048, α =1.07, and β =0.32. With probability p, δ(0) was chosen and w =0 was returned. With probability 1-p, the gamma (α,β) density was chosen and a value, w, was returned using the procedure of Schmeiser and Lal (1980) in the IMSL routine RNGAM. The returned w was converted to an efficiency score, v, according to v =exp{w0.5}. For each v, a vector Y was generated on the convex polytope with extreme points e1 =(v/a1*,0,0,0), e2 =(0,v/a2*,0,0), e3 =(0,0,v/a3*,0) and e4 =(0,0,0,v/a4*) using the method given in Devroye (1986).

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Data Mining: Opportunities and Challenges
ISBN: 1591400511
EAN: 2147483647
Year: 2003
Pages: 194
Authors: John Wang